Characterizing short-term stability for Boolean networks over any distribution of transfer functions

Abstract

We present a characterization of short-term stability of Kauffman's N-K (random) Boolean networks under arbitrary distributions of transfer functions. Given such a Boolean network where each transfer function is drawn from the same distribution, we present a formula that determines whether short-term chaos (damage spreading) will happen. Our main technical tool which enables the formal proof of this formula is the Fourier analysis of Boolean functions, which describes such functions as multilinear polynomials over the inputs. Numerical simulations on mixtures of threshold functions and nested canalyzing functions demonstrate the formula's correctness. To the best of our knowledge, previous formal characterizations only worked for special cases of "balanced" families.

Phase transitions and assortativity in models of gene regulatory networks evolved under different selection processes

“…Furthermore, these networks are often thought to exist near some critical point [2][3][4], where dynamic variability is maximized without reaching widespread network failure/breakdown. The phase transition between stability and instability in networks has been widely investigated, with studies focusing on the effects of topological features [5][6][7][8], dynamical features [9][10][11][12] or both [13] and their contributions to the location and behaviour of the transition. Of particular interest is the evolutionary process that leads to this critical point and how this process depends on both the topological and dynamical properties of the network and its nodes.…”